What can kinetic Monte Carlo do for active Matter?

As an efficient numerical method, discrete-time, continuous-space Monte Carlo (MC) is wildly used in physics. While constructing an active matter version is straightforward, the question remains to what extent it faithfully captures real-world active systems. We focus on a kinetic MC version for the simplest kind of active matter: persistently moving, non-polar, interacting particles. On the multi-particle level, the MC dynamics captures not only Motility-induced phase separation (a characteristic active matter phenomenon) but also features a non-equilibrium extension of the celebrated two-dimensional melting. An attempt to characterize these phases and their transitions traditionally relies on the existence of a thermodynamic pressure, which is not guaranteed outside equilibrium. In fact, small modifications of the MC rules have existential consequences for thermodynamic pressure. For a soundly chosen version of the MC dynamics, we show that pressure is a thermodynamic state variable over a robust parameter range. This is demonstrated by deriving the corresponding Langevin description and the associated expression for pressure, which is confirmed by large scale many-particle simulations. This work, therefore, contributes to equipping a MC toolbox for active Matter.